What is the definition of sine in a right triangle?

In a right triangle, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. For example, if angle A is opposite side a and the hypotenuse is c, then sin(A) = a/c.

How do you find the cosine of an angle in a right triangle?

The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the length of the hypotenuse. If angle B is adjacent to side b and the hypotenuse is c, then cos(B) = b/c.

What is the Pythagorean theorem and how is it related to trigonometry?

If sin(θ) = 0.6, what is the length of the opposite side if the hypotenuse is 10?

Hint: Use the sine definition.

Using the definition of sine, sin(θ) = opposite/hypotenuse. Therefore, 0.6 = opposite/10. To find the opposite side, multiply both sides by 10: opposite = 0.6 * 10 = 6.

For what angle θ is cos(θ) = 0.5?

Hint: Think of special angles.

The cosine of θ equals 0.5 at two angles within the range of 0° to 360°: θ = 60° and θ = 300° (or θ = π/3 and θ = 5π/3 radians).

What are the trigonometric ratios for a 45° angle?

For a 45° angle, both sine and cosine are equal: sin(45°) = cos(45°) = √2/2, and tangent is 1: tan(45°) = 1.

What is the sine of 90°?

The sine of 90° is equal to 1: sin(90°) = 1.

What is the cosine of 0°?

The cosine of 0° is equal to 1: cos(0°) = 1.

What is the relationship between sine and cosine for complementary angles?

Hint: Use complementary angle identities.

For complementary angles A and B, the relationship is: sin(90° - A) = cos(A) and cos(90° - A) = sin(A).

What is the tangent of an angle in terms of sine and cosine?

Hint: Recall the definitions of the functions.

The tangent of an angle θ is defined as the ratio of the sine of the angle to the cosine of the angle: tan(θ) = sin(θ)/cos(θ).

If cos(θ) = 0.8 and θ is in quadrant I, what is sin(θ)?

Hint: Use the Pythagorean identity.

Using the Pythagorean identity, sin²(θ) + cos²(θ) = 1. Therefore, sin²(θ) = 1 - (0.8)² = 1 - 0.64 = 0.36, giving sin(θ) = √0.36 = 0.6.

What is the maximum value of the expression 3sin(θ) + 4cos(θ)?

Hint: Consider the amplitude of the resulting sinusoidal function.

The maximum value occurs when sin(θ) and cos(θ) are at their optimal values. The maximum value is √(3² + 4²) = √25 = 5.

What is the double angle formula for sine?

Hint: Think about the sine addition formula.

The double angle formula for sine is: sin(2θ) = 2sin(θ)cos(θ).

If tan(θ) = 3/4, what is sin(θ) and cos(θ)?

Hint: Use the definitions of sine and cosine.

Using the definition of tangent, tan(θ) = opposite/adjacent = 3/4. The hypotenuse is √(3² + 4²) = √25 = 5. Thus, sin(θ) = 3/5 and cos(θ) = 4/5.

What is the law of sines?

Hint: Relate the sides of a triangle to its angles.

The law of sines states that in a triangle, the ratio of the length of a side to the sine of its opposite angle is constant: a/sin(A) = b/sin(B) = c/sin(C).

What is the area of a triangle given two sides and the included angle?

Hint: Use the formula involving sine.

The area A of a triangle can be calculated using the formula: A = (1/2)ab sin(C), where a and b are the lengths of two sides and C is the included angle.

What is the sine rule for finding angles?

Hint: Think about rearranging the law of sines.

The sine rule can be rearranged to find angles: sin(A)/a = sin(B)/b = sin(C)/c, allowing you to find an angle if you know the corresponding side lengths.

In a right triangle, if one angle is 30°, what are the lengths of the sides in relation to the hypotenuse?

Hint: Use known ratios for special angles.

In a 30-60-90 triangle, if the hypotenuse is h, then the side opposite the 30° angle is h/2 and the side opposite the 60° angle is h√3/2.